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Theorem sb4e 1877
Description: One direction of a simplified definition of substitution that unlike sb4 2006 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
sb4e  |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  E. y ph ) )

Proof of Theorem sb4e
StepHypRef Expression
1 sb1 1641 . 2  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
2 equs5e 1841 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
31, 2syl 15 1  |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  E. y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531   [wsb 1638
This theorem is referenced by:  hbsb2e  1995  hbsb2eNEW7  29518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535  df-sb 1639
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